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A MULTIVARIATE EXTENSION OF
FRIEDMAN'S X2-TEST
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University of North Carolina
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by
T. M. Gerig
Institute of Statistics Mimeo Series No. 550
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September 1967
Work supported by the National Institutes of
Health, Public Health Service, Research Grant
GM-12868
DEPARTMENT OF BIOSTATISTICS
-
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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A MULTIVARIATE EXTENSION OF
FRIEDMAN'S X2-TEST*
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by
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T. M. Gerig
Abstract
This paper deals with a multivariate extension of Friedman's
x;-test.
A rank permutation distribution and the large sample
properties of the criterion are studied.
The asymptotic relative
efficiency (A.R.E.) for a sequence of translation alternatives is
studied and bounds are given for certain special cases.
It is
shown that, under specified conditions, the A.R.E. of this test
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with respect to the likelihood ratio test is largest when the
block dispersion matrices differ and can be greater than unity
when the differences are large.
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1.
Suppose we are given p-variate data which form a complete two-way layout.
l
2
p
.
th
Let ~ij = (Xij,Xij,···,Xij) be the response from the plot in the i
th
that received the j
treatment; i=l,2, ••• ,n; j=1,2, ••• ,k; (k ~ 2).
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block
Assume
that the cumulative distribution function of ~ij is Fij(~)' ~£RP.
We wish to test the null hypothesis
(1.1)
for i=1,2, ••• ,n,
against translation type alternatives of the form
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INTRODUCTION
HA:
for
Fij (~)
j~1,2, •••
= Fi
,k and
(~-$Cj ) ,
i=l,~,
(1.2)
••• ,n,
* Work supported by the National Institutes of Health, Public Health Service
Grant GM-12868.
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2
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(a 12
j ,aj , ••• ,aj ) • Under this type of alternative, we can write the
null hypothesis as
where
~j
&:
(i.3)
H :
o
In standard parametric tests of this hypothesis, it is usually assumed
that
~ij • ~ + ftj + ~i + .tij ,
where
~, ~j
and
~i
and block effects.
are the constant vectors of mean effects, treatment effects
The vectors
~ij
are nk independent random vectors follow-
ing a. common multinormal distribution with zero mean vector and a common disr «cov(eij,e
s
s' ..
persion matrix t
ij »)· Several such t~sts are discussed in
&:
Anderson [1], in Rao [7], and elsewhere.
Frequently, some or all of these assumptions cannot be justified.
The
subject of this investigation is a non-parametric test which allows the assumptions of block additivity and multinormality to be relaxed.
posed the
xr2 -test
Friedman [5] pro-
for the univariate case and its asymptotic efficiency was
later studied by Elteren and Noether [4].
A class of non-parametric tests,
which do not require the multinormality or independence assumptions, has been
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proposed and studied by Sen [8].
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depth by Chatterjee [3] for k=p=2 and by Sen [8], will be presented in the
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A brief outline of the rank permutation. principle, which was studied in
next section.
2.
The
t~st
PERMUTATION ARGUMENT
to be proposed is based upon intrablock rankings.
Ranking will
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To illustrate the permutation argument, form the n matrices
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•
R •
",i
Each row of
~r
~i
, i=l,2, ••• ,n.
RP •••
il
will be some permutation of the numbers l,2, ••• ,k.
to be the matrix derived. from
~i
by permuting the columns in such a way
that the numbers l,2, ••• ,k appear in sequence in the first row.
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the set of matrices which are permutationally equivalent to
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the parent distributions, even under H.
a
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We say that
two matrices, A and B, are permutationally equivalent if A can be obtained
'"
'"
'"
from B by a finite number of permutations of the columns of B.
Let
S(~*i)
"'v
'V.
~r.
Hence,
be
S(~~)
contains kl elements.
The distribution of ~t over all its (kl)P-l realizations will depend upon
zation of
~r,
the distribution of
fact, if R E S(R*i)' then P{R
",0
'"i
'"
~i
=R
over
However, given a particular reali-
IS(R*i)' H }
'VO
'"
will be uniform under H.
o
S(~~)
=
In
l/kl, no matter what be the
0
parent distributions.
E S(~~), i=l,2, ••• ,n, then
Finally, if R
'VO
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Define
i
= R
'VO i
n
-
11"
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, i=l,2, ••• ,nIS(R~), i=l,2, ••• ,n; H }
a
'"
(2.1)
P{R = R Is (R*i)' H } = (1) n
'"i
",oi
'"
a
k!
because intrablock rankings are used and complete independence is assumed
from block to block.
We are.now in a position to select a test function which depends upon
R , i=l,2, ••• ,n.
'"i .
.
Such a test will be
\
completel~ specified
by the conditional
probability law (2.1) and, hence, will be a similar test of Ha •
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@n
Henceforth. we will let
3.
denote the probability law given by (2.1).
THE PERMUTATION TEST
Define T; • (l/n)Ei:lR~j for s=1.2 ••••• p; j=1.2 ••••• k •. Note that T;
depends upon n.
It is easily seen that
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and. after some essentially simple steps. that
o•
k' s
for i'l'i'
s'
{(k-l)/k}[Et=lRitRit - (k+l)2/4]. for i=i'. j=j'.
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Finally. we find that
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(3.1)
n
k s S'
2
where cr[s.s']{~*) = (l/n{k-l»Ei=lEt=lRitRit-{k(k+l) /4{k-l». and 0jj' is
the Kronecker delta.
Define E(R*) = «o[
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Again. we note that cr[
~
'\,
s.s
,](R*) depends upon n.
'\,
,](R*») and
s.s
'\,
Thus.
Var{T) = /:i (8) E(R*l = V{R*) say,
'\,
where ~
= « (Ojj' k-l) Ink»
as defined in [1. p. 347].
'\,
'\, '\,
'\, '\,
and ® denotes the 'direct product of the two matrices
If we assume that E(R*) is positive definite.-then
'\,
'"
.
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Finally, using the well-known formula
where u' is (lxr) and A is (rxr), it follows that
'V
'V
21
12
• n
11 ••• 2
Now, for our test function we take the quadratic form
~ .. (T-T)'V-l(R*) (T-T) .
n
where T(lxp(k-l»
=
'"
'V 'V
'V
'V
(3.2)
'" '" '
«k+l)/2, ••• ,(k+l)/2).
After some simple steps, we
obtain
(3.3)
[s s']
where «a'
.
(R*») = «a[
'"
s ,s
,](R*»)
-1
'"
= E- 1 (R*).
'"
'"
It will be demonstrated later that, for large n and with some mild
restrictions on Fi , i=l,2, ••• ,n, E(R*) is positive definite with high probability.
"''''
If, in practice, it is found to be singular, then elimination of
the proper variables will remedy the problem.
Following the arguments of Section 2, we can now compute the exact permutation distribution of
1.n
and thereby, determine a randomized test of H which
0
is strictly distribution-free and has exact size a.
are large, these computations may be too
\
length~y
Howeyer, when p, k and n
to be practical.
In the
next section, we will study the asymptotic permutation distribution of ~-.
n
This
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will provide us with a large sample approximation to the permutation distribution of
tn
and, thereby, increase the usefulness of the test.
4.
ASYMPTOTIC PERMUTATION DISTRIBUTION OF
tN
Define Fij[sl(x) and Fij[s,s'l(x,y) to be the marginal cdf's of X~j and
•.8
s'
(Aij,X
) for s,s'=1,2, ••• ,p; i=1,2, ••• ,n; j=1,2, ••• ,k.
ij
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kk
+ L~ L~
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f f
00
j~t=l
for
s~s'=1,2, •••
-00
00
Fit[s s'l(x,y)dFij[s s'l(X'y)
~,
-
k(k~l)2 }
' -
,p; i=1,2, ••• ,n
• k(k+l)/12 for s=s'=1,2, ••• ,p; i=1,2, ••• ,n.
(4.1)
r(F i ) = «o[ s,s 'l(F
»), s,s'=1,2, ••• ,p~
~i 0
(4.2)
Let
~ ~ 0
Theorem 4.1:
If Fij is continuous for i=l,2, •••• ni j=1.2 ••••• k. then
E(R*)-(l/n)Einl{r(F- )}
i
= ~~o
~~
converges in probability to the null matrix-as n+OO.
Proof:
Define the function
1
~(X,y)
for
x > y
=
(4.3)
o
for
x oS. y.
k s s'
=
[l/(k-l)lEj=lRijRij - k(k+l)2/4(k-l), then O[ s,s 'l(R*)
~
k ~(s
(l/n)Einlo[
Now using (4.3) we can write Rsij = 1+Etel
Xij,Xs)
it
= S,S 'l(R~).
/\,10
Let O[s,s'l(~~o)
=
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and, hence,
From (4.3), it follows that
00
_L Fit[s] (x)dFij [S](x),
for
0,
j~t
for j=t,
that
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and that, if s~s', then
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00
-L -L Fit[s](X)Fit,[s,](Y)dFij[S~S'](x,y),
00
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for
t~t', t~j, t'~j
00
_L ~ Fit[s,s,](x,y)dFij[,s'](x,y),
=
0,
. otherwise.
Using the above results, we can show that
.
- .
1
E{O~LS,s ,](R
*)} = k 1
tV i 0
-
~k
rrr
k k
;t~t'.l
00
f f
-00
00
-00
.
Fit [ ](X)Fit ,[ ,](y)dF i "[
'](x,y)
s
s
J s,s
j~t'
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••
00
+
k k
~
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f
f
j~t=l
-00
00
Fit[s s,){x,y)dFij[s , s'](x,y) -00'
k(k-l)J
4.
= O[s,s']({io)'
·J
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8
n
and hence, that E;{C1[
== (lIn) E
S,s ,](R*)}
~
'ULo
i ... l C1[ s,s ,](f.,).
s
Since R
is bounded above and below for s=l,2, •••,p; j=l,2, ••• ,k;
ij
i=l,2, ••• ,n, it follows that Var{C1[
••
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'](~~)}
has an upper bound that is
'ULO
Thus, (1/n2)Ein1 Var{C1[
,](Ri* )}~O as ntoo and so,
.
s,s
0
independent of i and n.
by the weak law of large numbers C1[
,
n
,](R*)-(l/n)Ei
1C1[
,](¥.,)
s,s
~
==
s,s
' o
U4
P
~
0 as
ntoo , which completes the proof.
Theorem 4.2:
n
Under H0 , (l/n)Ei == 1C1[ s,s ,](F
)
,
~i 0
= C1[ s,s ']'
where
(4.4)
C1[ s,s '] =
k(k+1)/12,
for s=s',
00
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s,s
Ti[S,S'] ... 4
Pgi[s,s'] = 12
00
f f
-00
-00
00
00
f f
-00
-00
[Fi[s,s,](x,Y)-~JdFi[s,s'J(x,y);
[Fi[s](X)Fi[s,](Y)-~]dFi[S,s'](x,y),
th
and where Fi[s](X), Fi[s,s'](x,y) are the sand (s,s')
th
margina1s of Fi(~)
given by (1. 2) •
Proof:
The theorem follows from Theorem 4.1 after setting
Fij(~)
=
Fi(~).
If F ,F 2 , ••• ,F are continuous and E = «C1[
']» is positive
1
n
~
s,s
~ s
k+1
definite, then the joint distribution of {n_(T - -Z-); s=l,2, ••• ,p;j=l,2, ••• ,k-1}
Theorem 4.3:
-
j
is asymptotically p(k-l)-variate normal with mean vector'zero and dispersion
matrix ~ ®.t where ~
Proof:
k
ra
j e 1 sj
=.
«0jj ,-11k» •
Let a j; s-1,2, ••• ,p; j=l,2, ••• ,k be
s
=0 for s=l,2, ••• ,p. Then write
'\
~ny
sp constants such that
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where
U1 m
Now, we find that E{Uil~n}
E{U~I<Pn}
=
m
! r aSjR~j'
sm1
j m1
0 and that
{f f r r
asjas'j
s=l s'=l j=l j'=1
,R~jR~j' ,16"'1
~
Ie
k(k+I)2}
4(k-l)
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=
{r
f' r
a j'a
s= 1 s '=I j =1 s s
'JO[
s,s ,](R*i)'
~ 0
Thus,
2
r
1
I.E{ Ui 1{P}=
r{taja'jlo[
,](R*).
n 1=1
n
s=l s'ml j=1 s s
s,s . ~
!
But,
_ k(k+1) <
(R*)
k(k+1)
12 - O[s,s'] ~ ~ 12
'
so that
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••I·
10
0=
O.
Thus, the conditions for the Liapounoff central limit theorem are satisfied, and we may conclude that Y is asymptotically normal.
n
An application
of some well-known results from multivariate analysis completes the proof.
Theorem 4.4:
Under the conditions of Theorem 4.3. the permutation distribu-
tion.of ~n is asymptotically chi-square with p(k-l) degrees of freedom.
Proof:
The theorem follows as an immediate consequence of Theorems 4.2 and
4.3.
Using Theorem 4.4,"we can now write the asymptotic permutation test of
Ho , given by (1.1), as
Reject H
o
if
Accept H
o
if
tn
> X2
"-p(k-l),a
2.
etn < X"-p(k-l),a,
-
(4.5)·
where
p{x 2 < X2
t -
t,a
}.
l-a, O<a<l.
We need to know conditions under which
we examine o[
s,s
t
is positive definite (pd).
If
']. in (4.4), we see that we can write
(4.6)
where Cl and Ci are constants, ~i • «Ti[s,s']»
and ~i • «Pgi[S,~']»'
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Since T and G are positive semi-definite, E will be pd when either terms
tV i
tV i
tV
on the right hand side of (4.6) is pd.
vector whose cdf is
••
I
then it can be shown that
1
p
matrix of (Fi[l](Xi), ••• ,Fi[p](Xi ».
~i
is the dispersion
Thus, ~i is pd unless there exists a
linear relationship between the variables Fi[S](~)' s=1,2, ••• ,p that holds
a.s.
Now, if there are m=m(n) values of i for which this relationship fails
to hold and if m(n)/n+A, where O<A<l as nt oo , then (l/n)Ei~~i is pd a.s. and
so is tV
Eo
Finally, we can show that the sequence of tests given by (4.5) is consistent, since,
1n -> n
s
ma x{T s _(k+1)/2}2/ cr[
](R*) = (12n/k(k+l»max{T - (k+l)/2}2 = 0 (n),
s, j j
s, S tV
.
S ,j
j
p
and so
p{tn -> Xp2 (k - 1) ,a. IH0 not true}+l as ntoo •
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Fi(~)'
1 2
If we let (Xi,xi,
••• ,XP ) be a random
i
5.
Theorem 5.1:
THE NON-NULL ASYMPTOTIC DISTRIBUTION OF {T;}
!f Fij is continuous for i=1.2 ••••• n; j=1,2 ••••• k, then
{n~(T; - ~;), j=1,2, ••• ,k; s=1,2, ••• ,p} is asymptotically normal with mean
vector zero and dispersion matrix
~
Ii r I [
i=l
t~t'=l
't* = «crj j '[s,s,]»,
cr*
II
where
•
j:;::::YldFit[S)(xldFit ·[s](Y)
_oo<y<x<oo
t~j, t '~j
+
If.
-oo<:x<y<oo
A~j(X'Y)dFit[S](X)dFit'[S](Y)]+ .
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12
1 n
I t Ik 1 [ JJ. AS1t(x,y)dF1j [ ] (x)dF, (x)
a*jj'[s,s] ... -n 1-1
_
=. ~x<y<oo
s
1j [s]
t;&j
t;&j ,
+
-
••
I
A~t(y,X)dF1j[ s ](x)dF1j' [s] (y)]
k
I [
AS (x,y)dF
( )dF
(
t=1 -oo<x<y<oo 1j
it[s] x
1j'[s] y)
t;&j
t;&j'
IJ
+
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JJ
~y<k<oo
II
-oo<y<x<oo
k
IJ
- tIl [
...
-oo<x<y<oo
t;&j
t;&j'
+
JI
-oo<y<x<oo
-11 A~j
00
A~ j (y,x)dFit [s] (x)dF1j' [s] ( Y)]
A~j,(x,Y)dF1j[ S ] (x)dF.1t[S] (y)
A~j,(y,X)dF1j[ S ](x)dF1t[s] (y)]
1
00
(x,y)dFij' [s] (x)dFij' [s] (YJ for
1n{k
i~1 t~1 ~ ~ B1~
a*
- -
00
jj[s,s'] - n
00
s s'
(x,y)dF1j [s,s'](x,y)
1
t;&j
+ Ik
t=1
ok·
00
00
t'~I~' ~
j+j'.
BS's'
1j (x,y)dFit[s](x)dFit,[s](Y)
t;&j t ';&j
.
for s;&s',
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00
00
- / /
,
B~j~ (x,y)dFij[S S,](x,y)
_00 -00
'
for
srJs' and j=lj",
where
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Proof:
Let a s j' s=1,2, ••• ,p; j=1,2, •••
. ,k be pk arbitrary constants.
Write
where
Now, it is easily seen that EUi-O and that
EU~ .
s
R J?
! ..!
s-l s'=l
k
I
j-l
k
,
raja ""COV{R~j,R~j-'} •
j '-I s s ....
Since R , i=1,2, ••• ,n; s-1,2, ••• ,p, j-l,2, ••• ,k are bounded above and below
ij
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s
s'
by 1 aad k, there exists upper and lower bounds for COV{Rij,R ij ,} which do
n
not depend upon i, j, j', s, s' nor n.
Thus,
r
E{U~} = O(n) and similarly
i=l
lim E n E{lu .I~}/[E n E{U 2 }]3/2
i=l
i
i=l
i
n~
= O.
Hence, by the LiapoUnoff central limit theorem and some well-known results
s s
from multivariate analysis, the asymptotic normality of {n~ (Tj-~j)}
is
established.
Finally, we obtain 0jj'[s,s'] • n
Using (4.3), it can be shown that
s
s'
COV{Rij,Rij ,} •
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(5.1)
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After some very tedious calculations, the first term on the right hand
side of 5.1 can be evaluated, and the expressions for 0jj'[s,s']' given in
the theorem, can be obtained.
This completes the proof.
It is useful to note that, under Ho (1.1), we find that 0jj'[s,s'] =
(8 jj ,-1/k)0[s,s'] where O[s,s'] is given by (4.4).
The rank of the asymptotic multinormal distribution can be at most p(k-l),
k
since
r
.
S
T = k(k+l)/2 for s=1,2, ••• ,p. In fact, when H · (1.1) holds and
j
o
j=
l.
\
F ,F , ••• ,F are continuous and are such that ~.is pd, the rank is p(k-l) and
n
1 2
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(5.2)
where
«a [s ' s'] »
= «a[s,s']»
-1
,is asymptotically chi-square with p(k-l)
degrees of freedom.
6. ASYMPTOTIC DISTRIBUTIONS OF $..n AND 1*n
UNDER A SEQUENCE OF TRANSLATION ALTERNATIVES
For the purpose of studying asymptotic power efficiency, we shall concern ourselves with the sequence of translation alternatives, {Hn }, given by
s
s
where xnj - x
fails to hold.
Throughout. this section, we will
write~:f
to mean that -F1 ,F 2 , ••• ,Fn
are absolutely continuous and are such that f\,t is pd.
We note that if ~1-~2-... ~,then Fij(~) - Fi(~) for i-l,2, ••• ,n;
j-l,2, ••• ,k.
Theorem 6.1:
If
-~ s s
{E1, then, under {Hn }, - {n
(Tj-~j)j j=1,2, ••• ,kj s=1,2, ••• ,p}
is asymptotically normal with mean vector zero and dispersion .matrix !J. (8)
tV
where
~
• «Ojj,-l/k»
Proof:
and
~
r
tV
is given in Theorem 4.3.
Using 6.1, Taylor's theorem and some routine analysis, the
theorem follows from
Theorem 6.2:
Theore~5.1.
If FE~, then,- under {Hn }, the asymptotic
distribution of
-
-tV
~n~ defined by (3.3), is
non-central chi-square with p(k-l) degrees of
freedom and non-centrality parameter
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r r q[S,S']ASAs '
(6.2)
s=l s'=l
where
AS
=
(k/n)E i : 1
f
co
[fi[s](x)]2dX, s=1,2, ••• ,p,
-co
(6.3)
y
f
fi[s](x)dx = Fi[s](y), and
-co
Proof:
[S s']
«a '
»
«a(s,s']»
=
-1
•
Under {H land FEf, it can be shown that (1/n)E ' n {E(F )} - E
n
I\,
i =1 I\, I\, i 0
I\,
tends to the null matrix as nt co •
Thus, we have, by Theorem 4.1, that E(R*)-E
I\,'V
converges in probability to the null matrix as ntco •
IE{n~(T;-(k+l)/2)}-a;Asl
7.
!
O.
'V
Also, under {H },
n
The theorem now follows from Theorem 6.1.
ASYMPTOTIC POWER EFFICIENCY OF
tn
For the purpose of studying asymptotic relative efficiency (A.R.E.), we
will confine our attention to the well-known likelihood ratio test (l.r.t.)
discussed in Anderson [1, pp. 187-210].
Unlike many of the parametric tests
of H , the non-null asymptotic properties of the l.r.t. are known.
o
In fact,
under the sequence of alternatives {H }given by (6.1), the l.r.t. statistic,
n
U , can be shown to be asymptotically non-central chi-square with p(k-l)
n
degrees of freedom and non-centrality parameter A2 = Ejkla~ X-la"
Un
= 'VJ I\, 'VJ
~j"
-'"
j=1,2, ••• ,k is given by (1.2),
Thus, "the A.R.E. of
tn
--1
~
-'"
= (n
-1
n
Ei :LAi)
= 'V
-1
where
, and A, • Var(x,,).
-
'VJ.
I\,J.J
with respect to U can be taken as
n
(7.1)
U ~ O.864k/(k+1)
n' n
uniformly in ~1,F2, •• ",Fn' for absolutely continuous Fi , i=1,2, ••• ,n, with
In the univariate case, Sen __ [9] has shown that
finite, non-zero variances.
e~
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es. U d epends upon CX", -, "'~t.. and AS, an d ,
n' n
"'.. ~
hence, a bound would be very difficult to establish.
I n th e mu ltivar i a t e
For the special case when ~i = A~ ~ and ~j has identically distributed
components, it is easily shown that eL u reduces to that of the univariate
n' n
case and, hence, the results given by Sen [9] apply.
Sen [9] studied the A.R.E. of Friedman's X:-test for the case Fi(x)=F(x/a ),
i
i-l,2, ••• ,n.
••I
These results may be extended to the multivariate case.
We have
s s'
2
k-l
A1. - Lj"l~j~ ~j' where ~ • «a[s,s,]/A A » and, under Fi[s](X)=F[s](x/Ai[s])'
n
we find that
where
CD
B k-L
S
•
[f[s](x)]2 dx and A~[s] • (~i)s,s.
2
2
,-1
--1 -1
2
Now, if ~i • ci~' c i > 0, then Ai[s] • ciA s and ~ - c ~ '. where As =
-
I
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I
I
cas~,
and c - n
-1
n
Li_lc i •
Thus,
where
But
and hence,
with equality
ditions,
holdi~g
only when cl.c2
cn~
Thus, under the stated con-
U is a minimum when c ..c
·c and can be made larger than
1 2
n
n' n
.
one by a suitable choice of c l ,c 2 ' ••• ,cn •
e~
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Turning attention to another aspect of interest, suppose we make the
assumption of normality and equal block dispersion matrices, then for p=2
we have
where
is the common dispersion matrix.
eLn,u
are obtained when 0jl
z
It can be shown that the extreme values of
0j2 for j z l,2, ••• ,k or when 0jl = -Oj2 for
n
j~1,2, •••
,k.
Thus
max
°jl,Oj2
j=1,2, ••• ,k
3k(1+p)/n(k+l) (l+p*),
for p
3k(1-p)/n(k+l) (l-p*),
for p 5.. 0,
3k(1-p)/n(k+l) (l-p*),
for p
3k(1+p)/n(k+l) (l+p*),
for p 5..
~
0
and
min
°jl,Oj2
j=1,2, ••• ,k
~
0
o.
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It is easily seen that, for fixed k, eM(p,k) is symmetric about p=O
and is concave for -1
~
P
~
0 and 0
3k/n(k+l) ~ eM(p,k) ~ .966.
~
p
~
It can be shown that
1.
The lower bound is obtained when Ipl~l or p=O.
3T-2 P ~ 1, .given in Kendall [6, p. 12], it can
g
be shown that eM(p,k) is monotonic increasing in k. In view of this, and
Using the inequality -1
~
max lim
p k~ eM(p,k) •• 966 and
after some
. numerical calculation, we find that
occurs when p-.52.
The following table gives the quantity
max
p eM(p,k) for
some values of k and gives the corresponding A.R.E. for the univariate x2-test.
r
2
3
4
5
6
7
10
20
100
.725
.790
.828
.852
.869
.881
.905
.934
.960
.966
p
.69
.68
.67
.67
.66
.66
.64
.61
.54
.52
e~,F
.637
.716
.764
.796
.819
.836
.868
.909
.945
.955
k
max
p
eM(p,k)
It is worth noting that the A.R.E. for X2 is smaller than that for
r
co
tn ,
under these assumptions.
Turning our attention to e (p,k), we see that, as a function of p, it
m
.
is symmetric about p-O. Also, we see that e (p,k) is monotinically decreasing
m
in Ipl and increasing in k.
o ~ em(p,k)
~
Finally, we find, by direct analysis that
3k/n(k+1) and that
II:"
lim
~3/4 ~ ~
em(p,k)
~
3/n.
The lower bounds
are obtained when Ipl~l and the upper when p-O.
In practice, having Ipl near
unity will lead to degenerate test statistics.
Hence, more meaningful bounds
can be obtained by restricting the range of p.
Chatterjee [3] has studied
the case p=k=2 so we will
~onfine
our attention to k
table gives values of Po and m(po) where m(po)
o ~ Jp I
~ Po and k ~ 3.
~
~
3.
em(p,k)
The following
~
3k/n(k+l), when
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Po
m(p )
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
.69
.67
.64
.61
.57
.52
.47
.40
.30
.00
These results follow from the stated monotonicity properties and some direct
calculations.
Results similar to these have been obtained by Bickel [2].
Finally, if we assume that
~ij
follows a p-variate normal distribution
with dispersion matrix
1 P
A
'"
=A
2
pIp
p
then e
u =
t n' n
3(1-p)/~(k+l)(1-p*)
P
P
1
and hence the bounds developed above may
be applied.
8.
NUMERICAL EXAMPLE
To illustrate the use of the test, bivariate normal data was generated
and recorded in a two-way table with k=3 and n=lO.
Then 1 was added to
each observation (both variables) in the second group and 2 was added to
each observation in the third group.
table.
The results appear in the following
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Groups
1
1
2
3
4
5
6
7
8
9
10
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Obs.
0.547
-0.575
1.706
1.252
-0.288
-0.310
1.417
0.932
0.878
0.819
-0.680
0.497
0.056
-0.285
0.711
0.089
-1.335
-0.349
1.635
0.845
2
Rks.
1
1
2
1
1
1
3
1
2
2
1
1
1
1
2
1
1
1
2
1
Obs.
1.811
1.840
2.509
1.574
2.524
1.553
0.703
1.390
0.094
0.045
2.077
1. 747
0.542
0.760
0.269
1.076
1.545
1.471
0.200
1.480
3
Rks.
2
2
3
2
2
3
1
2
1
1
2
3
2
2
1
3
2
2
1
2
Obs.
2.561
2.399
1.414
3.059
3.310
0.560
0.961
3.083
1.682
3.348
3.181
1.355
2.983
2.332
1.662
0.960
2.920
4.121
2.065
3.391
The parametric analysis gave Un - I~EI/I~E+~HI
that
[(1-U:)/u~][(c1-1)/c2] •
Rks.
3
3
1
3
3
2
2
3
3
3
3
2
3
3
3
2
3
3
3
3
.. 0.3083 and
it is known
Fo ' where c 1 • .(n-1) (k-1) and c 2 - (k-1), is
distributed as F2c ,2(c -1) when p-2.
1
2
significant at the .0005 level.
Thus, we find that F0 '. 6.807 and is
To compute ~ , we have
n
I;(R*) • [1
0.4
~ ~
and so
1n
so that
tn
~ext,
0.4] r-1(R*). _[ 1.1905
1 '~~
-0.4762
-0.4762]
1.1905
.. 10[(1.1905)(2.08)-(2)(0.4762)(0.79)] .. 17.238.
But
tn ~ ~(k-1)
is significant at the .005 level.
using the same bivariate data, the observations in each block
were multiplied by a constant.
In fact, the multiplier for block 1 was 1,
for block 2 was 1.2, for block 3 was 1.4, and so on.
added to the observations of groups 2 and 3 as before.
Then constants were
In this case, the
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parametric analysis gave U
n
= 0.4854
and F = 3.700 which is significant at
0
the 0.25 level, whereas the non-parametric analysis gave
tn
= 16.044
which is
significant at the .005 level.
These figures verify the results obtained in Section 7 pertaining to
ACKNOWLEDGMENT
The author is indebted to Professor P. K. Sen for suggesting this prob1em and for his help and guidance.
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REFERENCES
[1]
Anderson, T. W., An Introduction to Multivariate Analysis.
John Wiley and Sons, Inc., 1965.
New York:
[2]
Bickel, P. J., "On Some Asymptotically Nonparametric Competitors of
Rotelling's T2," The Annals of Mathematical Statistics, 36(1965),
160- 73.
.
[3]
Chatterjee, S. K., "A Bivariate Sign Test for Location,"
of Mathematical Statistics, 37(1966), 1771-82.
[4]
Van Elteren, P. H. and Noether, G. E., "The Asymptotic Efficiency of
the )(.-Test for a Balanced Incomplete Block Design," Biometrika,
46(19S9), 475-7.
[5]
Friedman, M., "The Use of Ranks to Avoid the Assumption of Normality
Implicit in the Analysis of Variance," Journal of the American
Statistical Association, 32(1937), 675-99.
[6]
Kendall, M. G., Rank Correlation Methods.
Company, Limited, 1955.
London: Charles Griffin and
[7]
Rao, C. R., Linear Statistical Inference.
Sons, Inc., 1965.
New York:
[8]
Sen, P. K., "On a Class of Nonparametric Tests for MANOVA in Two Way
Layouts," To appear in Sankhya.
[9 ]
Sen, P. K., "A Note on the Asymptotic Efficiency of Friedman's X2-Test "
r
'
To appear in Biometrika, 54(1967).
The Annals
John Wiley and